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Formation of field-reversed-configuration plasma with punctuated-betatron-orbit electrons

D. R. Welch,a S. A. Cohen,b T. C. Genoni,a A.H. Glasserc aVoss Scientific, Albuquerque, NM 87108 bPrinceton Plasma Physics Laboratory, Princeton, NJ 08544 cDept. of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195

submitted: January 6, 2010 revised: March 26, 2010

Abstract

We describe ab initio, self-consistent, 3D, fully electromagnetic numerical simulations of current drive and field-reversed-configuration plasma formation by odd-parity rotating magnetic fields (RMFo). Magnetic-separatrix formation and field reversal are attained from an initial mirror configuration. A population of punctuated-betatron-orbit electrons, generated by the RMFo, carries the majority of the field-normal azimuthal electrical current responsible for field reversal.

Appreciable current and plasma pressure exist outside the magnetic separatrix whose shape is modulated by the RMFo phase. The predicted plasma density and electron energy distribution compare favorably with RMFo experiments.

2

Electrical currents may be generated in magnetized plasma by a number of electrode-less methods. Inductive, radiofrequency-wave,1 energetic-beam-injection,2,3 and bootstrap4 techniques are widely used to drive currents parallel to the magnetic field while perpendicular currents may be generated by beam injection,5,6 diamagnetism, the thermoelectric effect,7 and rotating magnetic fields (RMF).8 The latter group is particularly relevant to field-reversed-configuration (FRC)9 plasma, see Figure 1a), unique among toroidal plasma in having only poloidal magnetic field and a magnetic null on the magnetic axis where the plasma energy density is highest. Producing current at the null is particularly difficult.7 This paper will provide physical insights, supported by detailed self-consistent calculations, into a novel non-resonant radio-frequency technique whose symmetry properties promote direct and efficient generation of field-normal currents in FRCs, even on-axis.

FRC-like plasmas are frequent in planetary and astrophysical settings.10 They are also created in the laboratory and therein used to study magnetic relaxation and reconnection processes,11 and stability and transport12,13 and to explore FRCs as potential fusion reactors.14,15,16 Many powerful methods of plasma theory, including magnetohydrodynamics (MHD), gyrofluid, drift-wave- instability,17 and even Taylor relaxation18 models are deficient for FRC plasma because of the distinctive FRC properties.19 Detailed understanding of and predictive capabilities for FRC plasma behavior require new theoretical tools.

RMFs, primarily of even parity (RMFe), have been used to form and sustain FRC plasmas and to heat their electrons. Odd-parity RMFs (RMFo) are predicted to perform the aforementioned and additional functions, such as heating ions,20 improving confinement21 and increasing stability.22

15,23 Recent experimental studies of RMFo have provided limited support for certain of these predictions. The new theoretical tools for FRCs must also properly treat RMFo, which, amongst other effects, removes axial symmetry and adds a new characteristic time scale, the rotational 3 period, placing even more stringent demands on a plasma model. Herein, we describe specific

24 reasons for a particle-in-cell (PIC) method for modeling the RMFo/FRC and results uncovered using it. One important result is that the electrical current is not predominantly carried by smoothly drifting or circulating particles but by electrons whose trajectories alternate between fast- circulating, higher energy (low collisionality, ν) betatron orbits and slowly drifting, lower energy

25 cyclotron orbits, with a net time-averaged azimuthal speed, 〈vφ,e〉, nearly equal to the RMFo’s.

This ratchet-like motion, see Figure 1b) and inset, which we term punctuated betatron orbits, has been observed in our earlier single-particle simulations20 but never before in a self-consistent simulation in which the effects of the full electron distribution function on resistivity, microinstability, and transport are included. Moreover, RMFo current drive does not depend on a wave-particle resonance central to the high efficiency method of Fisch26 hence can operate at a slow wave phase velocity and still generate high-energy low-ν electrons.

A distinguishing feature of RMFo is a time-varying azimuthal electric field, , generated near and on the plasma mid-plane (z = 0), which also contains the defining O-point field null line, the

FRC’s magnetic axis. Importantly, has both clockwise (CW) and counterclockwise (CCW) regions that rotate at ωR, the RMFo frequency, see Fig. 1b). Cyclotron-orbit electrons in the CW region E×B drift towards the null line, become betatron orbits, and then accelerate along the null.

These betatron-orbit electrons then enter the CCW region, decelerate, become cyclotron orbits and

slowly drift away from the null line, waiting for the RMFo to bring the CW "! region back to them to begin the ratchet-like azimuthal motion anew.

MHD or gyrokinetic models cannot properly treat a null or the direct and rapid -driven particle acceleration near the null line. These models are also far from adequate when either the ion or electron gyroradius, ρi,e, is a significant fraction of the separatrix radius, rs. Test-particle 4 techniques can accurately model ion or electron dynamics, even in the null region, but neglect the plasma response to the RMFo, such as whether the RMFo penetrates the plasma, if RMFo causes the magnetic-flux-surface shape to evolve, or whether turbulence develops and alters plasma dynamics.

By using PIC techniques, the present study avoids these deficiencies and provides the first fully self-consistent description of FRC formation from an initial low-β (ratio of plasma kinetic pressure to magnetic-field energy density) mirror plasma configuration.

For concreteness, we model a specific RMFo device, the Princeton Field-Reversed Configuration

(PFRC)15,27, sketched in Figure 1a). An 80-cm-long Pyrex cylinder is the vacuum vessel. Internal are 6 coaxial magnetic-flux-conserving copper rings (FC), three on each side of the midplane.

External to the Pyrex vessel and symmetric about its midplane is the RMFo antenna. Typical RMFo

characteristics are field strength BR ~ 10 G and frequency R / 2 =14 MHz. At an axial field at

the FRC's center of Ba =100 G, 90"ci ~ "R ~ "ce /20 , where !c = qBa / mc is the particle cyclotron frequency, m is the particle mass, q is the particle charge, and subscripts e and i refer to electron ! and ion, respectively.

A static mirror-configuration magnetic field is created by coaxial coils located near z = ±45 cm and z = ±105 cm. Nominally, these coils produce an initial axial bias field of strength Bo = 50 G at z = 0 cm and 2000 kG at z = ±45 cm. A necessary goal is for the RMFo to produce sufficient azimuthal plasma current to reverse the magnetic field at r = z = 0 cm. When this occurs, the field

(Ba = -Be) at the FRC’s center is about twice larger in magnitude than Bo. At the application of

RMFo power to the PFRC, the density rapidly rises. Within the first few µs, a near steady state is

12 -3 reached in which the plasma parameters are typically ne = 0.7-3 × 10 cm , Te =300-100 eV, and

Ti ∼1 eV. 5

28,29 PIC simulations, now described, were performed with the Large Scale Plasma (LSP) code.

LSP uses an explicit PIC algorithm, with standard particle-advance techniques augmented by a

30 novel energy-conserving push that avoids the so-called Debye-length numerical instability. LSP uses a temporally implicit, non-iterative, unconditionally stable electromagnetic field solver31 and a cloud-in-cell linear interpolation technique between particle locations and grid boundaries.

Approximately 200 particles per cell are used for each particle species.

The RMFo antennae are modeled with a sinusoidal current. The applied magnetic fields from the small- and large-bore coils at both ends of the PFRC are pre-calculated from a magnetostatic solution. Particles striking axial, radial and FC boundaries are removed from the simulation.

The spatial extent of the LSP simulation is r = {0,5} cm, = {0,2}, and z = {50,+50} cm, with grid spacings of Δ r = 0.15 cm, Δφ = π /4, and Δz = 0.2 cm. The explicit time-step limitation

"1 -11 6 requires #t < ! pe (∼ 10 s), corresponding to about 10 time steps. A typical simulation takes 4 days on a 32-processor cluster.

11 -3 A simulation begins with an ne ∼ 10 cm , Te = 4 eV hydrogen plasma seeded in the Pyrex vessel, along with room-temperature molecular hydrogen of density 3.5 × 1013 cm-3, corresponding to the PFRC fill pressure. The RMFo causes acceleration of plasma electrons and ionization of the

+ H2, hence plasma densification and electron heating. H2 is the dominant ion species formed in these relatively short simulations. Charged-particle collisions are treated using Spitzer rates.

Charged-neutral collisions are handled with a Monte-Carlo method utilizing energy-dependent

- tabular cross sections, σ. Scattering and ionization σ s for e -H2 from the literature are employed;

-15 2 σH2+--H2 is assigned a constant 10 cm . Neutral-neutral collisions assumed an isotropic scattering

-16 2 cross section of 7×10 cm . LSP calculates energy losses by collective radiation, charge exchange, 6 ionization, as well as conduction and convection to boundaries. Simulations are typically for 5 µs, during which time the neutral density drops about 1%.

Figure 2(a) shows the early time evolution of the total ion charge in the simulation volume for 5 values of the initial external field, labeled by Bo = Bz0 ≡ Bz(r,z,t) at t = r = z = 0. The amplitude of the RMFo field, BR, was 10 G. Positive Bz0 values correspond to the correct Bo direction to form an

FRC by the rotation sense of the RMFo. The figure shows density increasing exponentially with time with higher plasma densities attained at higher Bz0, doubling as Bzo is increased from 35 to 100

15 G), consistent with experiment. For positive Bz0, there is a temporary decrease in the rate of density rise between 600 and 900 ns, an effect we attribute to increased radial particle losses.

Density saturation occurs at about 5 µs, with the exact time depending on fill pressure, BR, etc. (For the Bz0 = 50 G, BR = 10 G case shown, the density at 5 µs is within 10% of that measured in the experiment.) When Bz0 is negative, the density rises slightly for 50 ns then decays. This critical simulation shows the importance of consistency between the sense of rotation and the initial Bo direction. Measurements on the PFRC have similarly shown a low density when Bz0 is negative.

Figure 2(b) shows the axial field strength versus time at z = 0 and r = 1 cm for the five values of

Bz0. For positive Bz0, the axial field strength falls with time and reverses for the lower Bz0 values.

The oscillations in Bz are at the RMFo frequency and are due to the proximity of the RMFo antenna's central arm. A fuller appreciation of field reversal can be gained from Figure 3 which presents snapshots in the r - z plane of three parameters, ne, Te, and Bz, at 5 times during the simulation with

Bz0 = 50 G. The top row shows ne. Though the total number of ions grows over the entire 2.5 µs period displayed, the radial location of the sharp density gradient shrinks between 50 and 375 ns and then grows until 1000 ns, by which time it reaches 3 cm. After t = 1 µs, the ne profile expands axially at a speed of 2.3 × 107 cm/s, about twice the ion acoustic speed. 7

The middle row shows Te (defined as 2/3 of the average electron energy) rapidly rising, reaching over 250 eV in isolated regions beginning at t ∼ 400 ns. For the next 500 ns, 50% variations in Te occur over 1-cm-scale -- comparable to ρe,i and c/ωpe -- axial and radial distances at a frequency above 200 MHz. (ωpe is the electron plasma frequency.) This turbulent period is concurrent with the aforementioned decrease in the density rate-of-rise and is coincident with a large value for the

drift parameter, γD = 〈vφ,e〉/ion thermal speed ~ 50. As ne continues to rise, Te becomes more homogenous, settling at about 125 eV at 1 µs. Electron energy fluctuations still occur at a reduced level, ca. ± 5 %. The Te profile inside the FC radius is nearly flat.

The bottom row shows the axial field. In the first 0.5 µs, little change occurs in Bz, but by t = 1

µs, a 50 % decrease is seen for r < 2 cm and |z| < 10 cm. At t = 1.5 µs, the azimuthal current has driven the central-region Bz to near zero. At t = 2.5 µs, field reversal is clearly evident in the region r < 1 cm, |z| < 8 cm.

Local projections of the magnetic field, i.e., contours of rˆBr + zˆBz (iron-filing plots), onto two orthogonal r - z planes at t = 2 µs are presented in Figure 4a). In both planes, a fully developed FRC is inferred, with O-point nulls at ro = 1.6 and 2 cm and rs = 3.1 cm, to be compared with rs = 1.9-3.0 cm reported in Ref. 15. The FRC shape strongly changes with RMFo phase, as predicted by Ref.

21. In conjunction with Figure 3, these data show a wide scrape-off layer and appreciable plasma pressure outside the separatrix. The changing shape of the separatrix and the oscillating position of the null repeat the intriguing question whether this dynamic variation in the plasma's shape may improve the configuration's stability against the internal tilt mode.22 Exploration of this question

will require far longer simulations and a different set of plasma parameters, e.g., higher rs"pi / Ec , lower ν, and lower BR/Ba. ! 8

ˆ Figure 4(b) shows an iron-filing plot of rˆBr +!B! in the r - φ plane at z = 7 cm, t = 2 µs. These local projections imply RMFo “penetration” to the FRC major axis. The field projections are twisted nearly 90° at r ∼ 2 cm, possibly by electron drag on the ions or, as we estimate, more likely

32 on the neutrals. For RMFe and the assumption of Spitzer resistivity, full penetration is predicted to occur when P ≡ γc/λ >2, where λ is the ratio of rs to the classical skin depth δ, and γc is the ratio of

ωR to ν. Including only electron-ion collisions P ~ 6. P falls to 1 adding electron-neutral collisions.

Figure 4c) shows 〈vφ,e〉/c versus radius for four axial positions, z = 0, 4, 8, 12 cm, ± 2 cm, at t =

2 µs: 〈vφ,e〉 ranges from 50 to 100% of the RMFo speed, ωRr, with electrons on axis and at larger radii having the higher percentage. Appreciable plasma current exists outside the separatrix because

of the high 〈vφ,e〉 and ne there. Inspection of 100’s of individual randomly selected super-particle trajectories from these PIC simulations show that punctuated betatron-orbit electrons contribute about 70% of the current for these low-s RMFo/FRCs.

The electron energy distribution function (EEDF) at t = 2 µs is shown in Figure 5. From 100 eV to 1 keV, the EEDF is well characterized by a single 120-eV exponential. A higher energy, ca. 180 eV, tail appears above 1 keV and is a far better fit to the experimental data than the Hamiltonian results15 which showed a sharp cut-off in the EEDF at ∼ 700 eV.

In summary, a 3D PIC plasma simulation technique has been applied to the study of FRC formation and electron heating by RMFo. While the net current flows smoothly, individual electrons responsible for the majority of the plasma current have a ratchet-like azimuthal motion, characterized by punctuated-betatron-orbit trajectories. This method of current drive has the potential for high efficiency because of the high energy (low ν) of the current-carrying particles.

Periods of large amplitude, high frequency, and short wavelength fluctuations in electron energy were observed and correlated with reduced density increase rate. The PIC results agreed well with 9 the measured plasma density, electron temperature, EEDF and separatrix location and also showed appreciable plasma pressure and azimuthal current outside the separatrix, whose shape was strongly modulated by both the flux conservers and the RMFo phase. These observations have strong ramifications for plasma transport and stability.

This work was supported, in part, by U.S. Department of Energy Contract No. DE-AC02-76-

CHO-3073. We acknowledge helpful discussions with Donald Voss and code assistance from

Robert Clark, Christopher Mostrom, and Clayton Myers. 10

Figure 1a). Cross-section of the PFRC device with field-reversed configuration magnetic field lines shown. Closed field lines, inside the separatrix, are red. The elongation, E, is the X-point location normalized to rs.

Figure 1b). An electron trajectory in the midplane of an RMFo-heated FRC, viewed along the major axis. The trajectory is a betatron orbit (blue), punctuated at beginning and end with counter-drifting cyclotron orbits (red). The RMFo-created azimuthal electric field, when the electron is midway in its betatron orbit, is shown (green). The O-point null line, as drawn, is an approximation since it neglects the RMF contribution. (Inset) Electron azimuthal position vs time. 11

Figure 2. a) Total ion charge in the simulation volume vs time, for 5 values of the initial axial field, Bz0,. b) Axial field strength at z = 0 and r = 1 cm vs time, for the same initial values of Bz0. 12

-3 Figure 3. (color online) Top row: log10 ne (cm ). Middle row: Te (eV). Bottom row: Bz (G). The five columns are snapshots at the following times, from left to right: 0.015, 0.5, 1, 1.5, and 2.5 µs. Color-contour scales are to the left.

Figure 4. (color online) At t = 2µs (a) Local projections of magnetic field lines in the r-z plane for two RMF phases, 90° apart. (b) Local projections of field lines at z = 7 cm in the r-φ plane. (c) Average electron azimuthal velocity at four z locations (±2 cm). The dashed line shows ωR r/c. 13

Figure 5. (color online) Calculated electron energy distribution functions (EEDF). The cases labeled 120 and 150 eV are Maxwellians; the LSP curve was taken at t = 2 µs into the simulation. 14

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